22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Surface Condensation and
Evaporation in Turbulent Flows
V. LÉAUTAUDa , Y. HOARAUb , R. MOSÉc
a. Laboratoire ICUBE - Université de Strasbourg, [email protected]
b. Laboratoire ICUBE, [email protected]
c. Laboratoire ICUBE, [email protected]
Résumé :
Dans le cadre du développement d’un système de bio-décontamination, la modélisation numérique est
essentielle pour s’assurer que la condensation s’effectue sur toutes les parois à purifier. Ce sytème devant
être adapté pour différentes conditions et échelles, le modèle CFD développé doit prévoir avec précision
les phénomènes physiques gouvernant ce type d’écoulement. Dans le cadre du projet ECOBIOCLEAN,
l’adaptation d’un code CFD avec l’implémentation des principaux mécanismes est expliquée, identifiés
comme la condensation et l’évaporation, la diffusion multi-composant, la décomposition chimique, les
transferts de chaleur et la turbulence. La validation du code est réalisée avec les cas de la littérature,
ainsi qu’une comparaison avec les résultats expérimentaux produits par un prototype disponible à la
Faculté de Pharmacie de Strasbourg.
Abstract :
In the context of the development of a bio-decontamination system, numerical modelling is required to
ensure that condensation occurs on all of the volume to purify. Because this system must be adapted
for various conditions and scales, the developped CFD model must accurately predict the physical mechanisms governing this type of flow. In the context of the ECOBIOCLEAN project, the adaptation of a
CFD code with the implementation of the main mechanisms is explained, identified as condensation and
evaporation, multicomponent diffusion, chemical decomposition, heat transfer and turbulence. The validation of the code is conducted in accordance to the literature and a comparison with the experimental
results gathered with a prototype available at the Faculty of Pharmacy of Strasbourg is performed.
Mots clefs : CFD, évaporation, condensation, diffusion.
1
Introduction
A bio-decontamination system is being developped and a prototype has been built to test the conditions
required to purify a volume. Since this system will be adapted for various applications of differnet scales,
the model has to be able to correctly model the system for any geometry and dimensions, from a small
test volume to a full scale clean room. Computational Fluid Dynamics, or CFD, is the most adaptable
and accurate way to predict the behavior of the fluids.
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
The system consists in a heater and an injector in a circuit which can be closed or opened.
During the first phase, the decontamination phase, a solution of hydrogen peroxide is sprayed in a hot
air stream. The droplets evaporate quickly, before reaching the volume to be decontaminated. There,
condensation begins to form on the cold walls. The circuit is closed in order to conserve the hydrogen
peroxide, to minimize the amount of solution to be consummed.
When all of the walls are covered with condensed hydrogen peroxide, the second phase begins, during
which hot and dry air is blown in the volume. The circuit is now opened, letting out the vapor not needed
anymore. This lasts until all of the hydrogen peroxide has evaporated.
2
CFD Model
2.1
Presentation of the solver
The CFD code used in the present work is called NSIBM, standing for Navier-Stokes Immersed Boundary Method. It is an massively parrallel incompressible Navier-Stokes solver developped at the ICube
Laboratory, Strasbourg, France. It uses unstructured cartesian meshes, where the boundaries that cannot
be modelled using only the cartesian mesh are modelled using the Immersed Boundary Method with
an automatic mesh refinement along the boundary. The implemented schemes include the 1st and 2nd
order for time and 1st to 3rd (TVD) for space. Turbulence can be modeled by Spalart-Allmaras, k- and
k-ω models.
The code has been validated using standard cases from the literature, such as the driven cavity [1] and
the buoyancy driven cavity [9] cases.
2.2
Model implementation
Modelling the decontamination system requires the implementation of several models, amongst which
are the condensation and evaporation rate, multicomponent transport diffusion, chemical decomposition
and the heat transfer.
— Vapor transport and diffusion : initially, there is no vapor in the volume. It is injected at the inlet.
The driving parameter is the total mass of solution sprayed in the volume. It is sprayed at a certain
rate until the saturation is reached.
— Condensation and evaporation happen on the walls. It can be expressed as a boundary condition.
Different laws have been suggested by many authors to model the condensation or evaporation
rate. The driving parameters are the vapor pressure and the temperature of the vapor close to the
wall, and the saturation vapor pressure at the boundary which is expressed as a function of the
condensation temperature.
— Heat transfer. Modelling condensation requires an accurate modelling of the temperature difference between the liquid, on the wall, and the vapor. This requires an accurate modelling of the
heat transfer of the condensation to the wall (and the outside) and to the gas stream.
— Chemical decomposition. One of the main advantage of using hydrogen peroxide for decontamination is its decomposition into water and oxygen, two friendly chemical components to the
atmosphere and the human. With increasing temperature, the reaction rate will increase. This is
not wanted during the first phase, since decontamination requires the hydrogen peroxide, but is
during the second phase (rince cycle).
22ème Congrès Français de Mécanique
2.2.1
Lyon, 24 au 28 Août 2015
Vapor transport
The assumption of quickly evaporating droplets leading to a single-phase flow, the problem only of mass
transfer revolves around the diffusion of the condensables in the flow.
Each component k is represented by its volume fraction αk , its molar fraction xk or its mass fraction ωk
in the flow. The perfect gas assumption leads to having the following equality : αk = xk .
The general multiphase Navier-Stokes mass balance equation for incompressible flows writes :
−
→ Γk
−
→
∂αk
1
+ ∇ · (αk Vk ) = − ∇ · Jk +
∂t
ρk
ρk
(1)
where Jk is the diffusion flux of the vapor in the mixture. This flux can be divided into several diffusion
fluxes, each accounting for a diffusion cause, molecular diffusion, turbulence induced diffusion and
thermal diffusion :
−
−−−
→ −−−→ −−−→ −−→
Jk,mass = Jk,mol + Jk,turb + Jk,th
(2)
In binary mixtures, the molecular diffusion flux in terms of mass fraction gradient are commonly defined
as :
−−−→
−−→
Ji,mol = −ρDi,j ∇ωi
Mi Mj −−→
= −ρDi,j
2 ∇xi
M
(3)
(4)
For the multi-component mixtures, the model uses a mixture-averaged diffusion coefficient, based on
calculations from Kee et al. [3] and Sutton and Gnoffo [7] :
−−−→
−−→
Using mass fraction Ji,mol = −ρDiMix ∇ωi
−−−→
Mi 1 − ωi −−→
or, using mole fraction, Ji,mol = −ρDiMix
∇xi
1 − xi
Pn M
j6=i xj Mj
with DiMix =
Pn
M j6=i xj /Dji
(5)
(6)
(7)
The turbulence adds an effect of diffusion. It is often caracterised by a turbulent Schmidt number. Similarly to molecular diffusion, the turbulent Schmidt number is defined as :
Sct =
µt
ρDt
(8)
where µt is the eddy viscosity. This leads to the expression of the turbulent diffusion coefficient, which
is directly used to model the turbulent diffusion :
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
−−−→
−−→
Ji,turb = −ρDt ∇ωi
µt
Dt =
ρSct
(9)
(10)
Tominaga and Stathopoulos [8] reviewed the studies regarding the turbulent Schmidt number used in
the literature. Mainly, the values of the turbulent Schmidt number used are 0.7 or 0.9. However, in a
significant number of cases, experimental values can go from 0.2 to 1.3, depending on flow parameters
and the geometry.
2.2.2
Condensation and evaporation models
It is the main mode of condensation of this study. This mode happens because the hot gas stream encouters the cold wall to which transfers heat and thus, cools down. The condensables phase then changes
on the walls, at a rate depending on several parameters.
Herz-Knudsen-Schrage equation :
equation, defined as :
2
ṁv =
·
2 − KC
M
2πR
Marek and Straub [5] suggest using the Herz-Knudsen-Schrage
#
1/2 "
pv
psat (Tl )
· KC · 1/2 − KE ·
1/2
Tv
Tl
(11)
where KC and KE are respectively the condensation and the evaporation coefficients. They are defined
as follows :
number of gaseous molecules absorbed by the liquid phase
number of gaseous molecules impinging the liquid phase
number of liquid molecules absorbed by the gaseous phase
KE =
number of liquid molecules emitted by the liquid phase
KC =
(12)
(13)
This equation has been used in many studies and is not subject to approximation as it does not need data
about the heat transfer coefficient, contrary to the following correlations. It also has the advantage of
being a reversible equation, meaning that this can be used to account for condensation and evaporation.
Heat and Mass Transfer Analogy : Another way of estimating the condensation is suggested by Liu
et al. [4] who studied the condensation in a building. They used the Lewis equation, and assumed a
constant heat transfer at the boundaries, directly linking the heat transfer rate to the mass transfer rate.
It will be refered to as the Heat and Mass Transfer Analogy (HMTA) :
hT −2/3
psat (Tl ) Tv
ṁv =
Le v
xv −
cp
p
Tl
Sc
α
where Le =
=
Pr
D
(14)
(15)
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Both condensation models will be studied and compared to the results obtained with the prototype of
the bio-decontamination system.
2.2.3
Wall heat transfer
According to all of the studies regarding condensation, the temperature difference between the bulk gas
and the liquid phase is a leading parameter in condensation. Therefore, an accurate modelling of heat
transfer is crucial to the study of condensation.
The temperature distribution in the bulk flow is calculated accurately with the classical convectiondiffusion equation.
If the condensation is represented by a continuous film on the walls, the heat transfer model is simplified.
In this case, the condensation can be represented by an extra resistance between the bulk flow and the
outside. The following thermal/electrical anology can be made to analyze the temperature profile. It is
assumed that there is no convection resistance between the liquid film and the wall. Also, due to the
phase change, an amount of heat corresponding to the latent heat of vaporization is released during
condensation. Herrantz et al. [2] suggests adding this mechanism in parallel to the convection.
Liquid film
Gas
Wall
Outside
R3
R4
Condensation
Rcond
Wall conduction
Text
b
Droplet conduction
b
b
R1f
Tw
R2f
b
Tl
b
Tg
Convection
Convection
Gas
Gas/liquid interface
Bulk/Wall interface Wall/Outside interface
Outside
Figure 1 – Thermal/electrical analogy of the wall heat transfer for filmwise condensation.
The heat flux can thus be expressed as follows :
q = hT (Tg − Text ) =
Tg − Text
R1f Rcond
+ R2f + R3 + R4
R1f + Rcond
(16)
Condensation heat transfer
Condensation generates energy, which was stored in the gas as latent heat of vaporization. This heat is
released during condensation.
qcond = hcond (Tg − Ti ) = ṁLv
(17)
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Using the Sherwood number formulation, the heat transfer coefficient expression is

Pv Psat
−
Tg
Ti 
Sh Mv Lv D 

=
·

L
R
Tg − Ti 

hcond =
1
Rcond
(18)
Herrantz et al. [2], based on the work of Peterson et al. [6] relative to the diffusion layer theory, suggest
making the approximation (19), using the Clapeyron equation (20) and making another approximation
for the specific volume change (21).
Pv Psat
1
−
≈
(Pv − Psat )
Tg
Ti
Tavg
∆P
Lv
=
∆T
T vfg
RTi
RTi
vfg = vg − vl =
− vl ≈
P v Mv
Pv Mv
(19)
(20)
(21)
After developping, the following expression is obtained :
hcond =
Sh Pv Mv2 L2v D
Φ
L R2 Tg Ti
(22)
where Φ is the ratio between the log mean molar fractions of vapor and noncondensables in the boundary
layer. This factor is added to correct the result with empirical observations on the effect of condensable
concentration.
Φ=
xv,avg
xnc,avg
1 − xnc,b
ln
1 − xnc,i
=−
xnc,b
ln
xnc,i
(23)
Convection between the wall and the exterior
The wall is passively cooled down with ambient temperature due to free convection with the exterior. The
McAdams correlation, valid for Rayleigh numbers in the range of 109 < Ra < 1012 and independant
of the characteristic length, is used :
Nu = 0.13Gr 1/3 Pr 1/3
1/3
gρext Cp λ2ext |Tw − Text |
1
= 0.13
⇒h4 =
R4
νext Text
(24)
(25)
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Nomenclature
Constants and Variables
Symbol
D
D
h
J
L
Lv
M
ṁ
P, p
q
R
R
v
V
x
λ
ρ
ω
Description
Diffusion coefficient
Binary diffusion coefficient
Heat transfer coefficient
Diffusion flux
Characteristic length
Latent heat of vaporization
Molar mass
Condensation mass flux
Pressure, partial pressure
Heat flux per unit surface
Thermal resistance
Universal gas constant
Specific volume
Velocity
Mole fraction
Thermal conductivity
Density
Mass fraction
Unit
m2 .s−1
m2 .s−1
W.K −1 .m−2
m
J.kg −1
kg.mol−1
kg.s−1 .m−2
Pa
W.m−2
(W.K −1 .m−2 )−1
J.K −1 .mol−1
m3 .kg −1
m.s−1
W.K −1 .m−1
kg.m−3
Subscripts and Superscripts
Symbol
avg
b
ext
f
g
i, j, k
l
m
nc
sat
t
T
v
w
Description
Average
Bulk
Exterior, outside
Film
Gas
Variable integer subscript
Liquid
Mass
Non-condensable gas
Saturation
Turbulence
Thermal
Vapor, condensable
Wall
Dimensionless numbers
Symbol
Le
Nu
Pr
Re
Sc
Description
Lewis number
Nusselt number
Prandtl number
Reynolds number
Schmidt number
Références
[1] U.K.N.G. Ghia, K. N. Ghia, and C.T. Shin. High-re solutions for incompressible flow using the
navier-stokes equations and a multigrid method. Journal of computational physics, 48(3) :387–411,
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
1982.
[2] L.E. Herrantz, M.H. Anderson, and M.L. Corradini. A diffusion layer model for steam condensation
within the ap600 containment. Nuclear Engineering and Design, 183 :133–150, 1998.
[3] R.J. Kee, G. Dixon-Lewis, J. Warnatz, M.E. Coltrin, and J.A. Miller. A Fortran computer code
package for the evaluation of gas-phase, multicomponent transport properties. Sandia National
Laboratories Livermore, CA, 1998.
[4] J. Liu, H. Aizawa, and H. Yoshino. Cfd prediction of surface condensation on walls and its experimental validation. Building and Environment, (39) :905–911, 2004.
[5] R. Marek and J. Straub. Analysis of the evaporation coefficient and the condensation coefficient of
water. International Journal of Heat and Mass Transfer, (44) :39–53, 2001.
[6] PF Peterson, VE Schrock, and T Kageyama. Diffusion layer theory for turbulent vapor condensation
with noncondensable gases. Journal of Heat Transfer, 115(4) :998–1003, 1993.
[7] K. Sutton and P.A. Gnoffo. Multi-component diffusion with application to computational aerothermodynamics. In 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, 1998.
[8] Y. Tominaga and T. Stathopoulos. Turbulent schmidt numbers for cfd analysis with various types of
flowfield. Atmospheric Environment, 41 :8091–8099, 2007.
[9] C. Wan, BSV Patnaik, and D GW Wei. A new benchmark quality solution for the buoyancy-driven
cavity by discrete singular convolution. Numerical Heat Transfer : Part B : Fundamentals, 40(3) :
199–228, 2001.